Welcome, Guest. Please Login.
Gravity Simulator
12/17/17 at 11:22:06
News: Registration for new users has been disabled to discourage spam. If you would like to join the forum please send me an email with your desired screen name to tony at gravitysimulator dot com.
Home Help Search Login


Pages: 1 2 
Send Topic Print
Libration on tidelocked planets (Read 8219 times)
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Libration on tidelocked planets
09/15/07 at 21:47:09
 
I posted this on BAUT, maybe someone here might be able to explain how this works...
 
------------
Usually, if you're on the surface a tidelocked planet orbiting a star in a perfectly circular orbit (and in a 1:1 spin-orbit resonance), you'd expect the sun to just be hanging in the same spot in the sky all the time (if you're on the side that can see it).
 
But if the planet's on an eccentric orbit, the star will move in the sky over the course of an orbit won't it? Obivously it'd get larger and smaller as the planet got to its perihelion and aphelion, but it should move laterally as well shouldn't it (I think that's down to the fact that the star is in one focus of the orbit)?
 
I was thinking that if you had the right circumstances on the ground then you could actually get false day and night cycles in the twilight zone as the star rose above and dipped below some mountains in the way during the orbit.
 
Taking a K4 V star (about the upper limit to where you can get a tidelocked planet in the habitable zone) with a luminosity of 0.16 sol and radius of 0.65 sol, the habitable zone should be at 0.4 AU. So the star's angular diameter 0.867 degrees, or almost exactly 52 arc seconds (1.62 times the angular diameter of the sun/full moon from earth).
 
So for this star at least it seems possible to be able to have the scenario I'm looking for - we'd only need the star to be able to move up and down by a degree or two over an orbit to be hidden behind mountains in the distance.
 
That being the case, is there a way to calculate exactly how far a primary star would move in the sky given a particular orbital eccentricity?
---------------
 
There's talk there of having to calculate it using Free/Mean/Eccentric anomalies, but I was wondering if there's any way to use Gravity Simulator to figure this out empyrically?
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Libration on tidelocked planets
Reply #1 - 09/16/07 at 00:32:00
 
I think theres not a formula for this ; maybe there's a formula for the maximum deviation ..
What you can do is simulate this in GravSim and output the longitude of the planet over one revolution .  
You'll get the longitude (0-360 degrees over 1 year) . In the second column you calculate the sine .  
Substracting both you'll get a periodic over time , giving you the difference .
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #2 - 09/16/07 at 02:15:12
 
I like Frank's idea.
 
You'll have 2 components to the libration:  The sun moving north and south of its average position, and the sun moving east and west of its average position.
 
The north/south movement will only apply to a planet whose axis of rotation differs from its orbital plane.  
 
Your planet's rotation will simply be linear.  If the planet is in a circular orbit, its true longitude will also be linear.  If you simulate a full orbit and output your data, you can figure out how far the true longitude deviates from linear.
 
Here's an image I made.  I used Gravity Simulator to create a planet around a star with eccentricity = 0.5.  I then computed its period and used the screen shot feature to take 12 screen shots every T=1/12 orbital period, with Trails = off.  I stacked the images together, along with a complete orbit with Trails = on.  Then I made a two-tone circle and pasted a copy of it on each of the 12 positions, rotating it 30 degrees (360/12) each time.  This helps illustrate the libration.

In reality, a planet with an eccentricity this high would probably be locked into a spin-coupled resonance like Mercury, but this exaggerated effect helps illustrate the libration.
 
For the North South libration, give your planet inclination that equals its axial tilt.  Then you could output the position vectors over a complete orbit and compute with the z-position how many degrees above or below the ecliptic the planet is.  That will tell you how far the star has drifted from its average position in its sky.
 
Maybe you could use some sine regression software to come up with a formula that fits your data.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Libration on tidelocked planets
Reply #3 - 09/16/07 at 02:18:38
 
"Maybe you could use some sine regression software to come up with a formula that fits your data. "
Such software is available ?
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #4 - 09/16/07 at 02:31:19
 
Quote from frankuitaalst on 09/16/07 at 02:18:38:
"Maybe you could use some sine regression software to come up with a formula that fits your data. "
Such software is available ?

I think so.  I think my calculator can do it.  Excel doesn't seem to have it built in.  Maybe IDL has a routine.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Libration on tidelocked planets
Reply #5 - 09/16/07 at 05:10:29
 
The formula may be simple ..., as it may not be needed  
Assuming that you are integrating a system as with GravSim you can calculate the planets period T from SMA a ( if not already given ) .  
You can also calculate the initial angle of the planet to its sun : arctg(( xp-xs)/r) this gives an initial value.  
You can calculate this angle again as the simulation runs . At time t the planet should have rotated over 360*t/T degrees . Subtract this value from the calculated value . This should give the apparent motion of the sun .  
As Tony points out you may also take into account the motion in NS direction.
Back to top
 
 
Email View Profile   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Libration on tidelocked planets
Reply #6 - 09/16/07 at 09:17:12
 
Thanks guys... I'll ponder on this. Though if the orbit isn't inclined, you'll only get east-west movement of the star in the sky, right?
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Libration on tidelocked planets
Reply #7 - 09/16/07 at 09:57:13
 
Right ...
BTW : the formula should be arcsin instead of arctg
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #8 - 09/16/07 at 13:24:12
 
I just noticed something interesting.  In my image, the 2 hemispheres do not represent the day/night side of the planet.  Rather, they represent a fixed hemisphere on the planet's surface.
 
But if you were to move the star to the empty focus in the ellipse, then the light blue side would represent the day side, as it always points to the empty focus.  I wonder if this is just coincidence, or if it works for all eccentricities.
Back to top
 
 
Email View Profile WWW   IP Logged
frankuitaalst
Ultimate Member
*****


Great site

Posts: 1507
Gender: male
Re: Libration on tidelocked planets
Reply #9 - 09/16/07 at 13:36:17
 
You're right !  
Is really amazing ! Is this geometrically correct if you draw lines to the focus ?
Back to top
 
 
Email View Profile   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #10 - 09/16/07 at 14:13:55
 
Quote from frankuitaalst on 09/16/07 at 13:36:17:
You're right !
Is really amazing ! Is this geometrically correct if you draw lines to the focus ?  

I printed the picture and tried that.  It's close but not perfect.  But I manually pieced together the image, so I wouldn't expect it to be perfect.
Back to top
 
 
Email View Profile WWW   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Libration on tidelocked planets
Reply #11 - 09/16/07 at 15:11:54
 
Quote from Tony on 09/16/07 at 13:24:12:
But if you were to move the star to the empty focus in the ellipse, then the light blue side would represent the day side, as it always points to the empty focus. I wonder if this is just coincidence, or if it works for all eccentricities.

 
It's true for all of them. The orbiting body always locks to the empty focus of the orbit, not to the primary object itself.
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #12 - 09/16/07 at 15:45:12
 
Quote from Mal on 09/16/07 at 15:11:54:
Quote from Tony on 09/16/07 at 13:24:12:
But if you were to move the star to the empty focus in the ellipse, then the light blue side would represent the day side, as it always points to the empty focus.  I wonder if this is just coincidence, or if it works for all eccentricities.  


It's true for all of them. The orbiting body always locks to the empty focus of the orbit, not to the primary object itself.

 
Neat.  Did you figure that out by verifying my claim, or was this already a well-know fact?
Back to top
 
 
Email View Profile WWW   IP Logged
EDG
Ultimate Member
*****


oh, crumbs!!!

Posts: 611
Gender: male
Re: Libration on tidelocked planets
Reply #13 - 09/16/07 at 16:33:41
 
Quote from Tony on 09/16/07 at 15:45:12:
Neat. Did you figure that out by verifying my claim, or was this already a well-know fact?

 
The latter. Look up synchronous rotation in the index of the SSD book (I think you said you had it?), the lock to the empty focus is mentioned on p44.
Back to top
 
 

(formerly known as Mal)
View Profile WWW   IP Logged
Tony
YaBB Administrator
*****




Posts: 1052
Gender: male
Re: Libration on tidelocked planets
Reply #14 - 09/16/07 at 16:40:24
 
Back to top
 
 
Email View Profile WWW   IP Logged
Pages: 1 2 
Send Topic Print